Martin Pavlov

We study a three-firm market following the breakdown of an initially complete cartel, taking both the breakdown and the identity of the departing firm as exogenously given. In the baseline model, the former cartel members no longer coordinate: all three firms choose quantities simultaneously and non-cooperatively. Thus, the term failed cartel describes the institutional origin of the post-cartel market rather than continued partial coordination or an endogenously profitable deviation. Because firms’ optimal quantities need not be unique, we represent their strategic behavior by set-valued best-response correspondences. We establish equilibrium existence for this simultaneous post-cartel Cournot model as a three-player specialization of the classical Debreu–Glicksberg–Fan framework, combining Berge’s maximum theorem with the Kakutani–Fan–Glicksberg fixed-point theorem. The generalized concavity of inverse demand and convexity of costs provide sufficient conditions for existence without requiring unique optimal responses. We also examine the single-valued case, distinguishing the uniqueness of individual best responses from the uniqueness of equilibrium. To enrich the benchmark analysis following the cartel breakdown, we introduce a separate anticipatory extension belonging to the class of hierarchical multi-leader–follower games. Our extension is narrowly focused on separating strategic from technological sources of production asymmetry. In our specialization, Firm 3 has a uniquely determined continuous response, while Firms 1 and 2 choose their quantities simultaneously and anticipate that response, yielding a two-leader–one-follower game. Equilibrium existence is established separately under explicit continuity and quasi-concavity assumptions on the induced payoffs. The extension shows that, even under identical technologies, the ordering of equilibrium quantities depends on the sign of Firm 3’s response to the leaders’ production decisions: depending on this response, either Firms 1 and 2 or Firm 3 may individually produce the larger equilibrium quantity. Thus, firms’ incentives to leave the cartel may differ according to the post-cartel market structure and their individual strategic positions within that structure.

Collusion is inherently unstable because individual firms have incentives to deviate from cooperative agreements. We study a novel hybrid oligopoly market structure, termed a failed cartel, in which one firm abandons a cartel agreement while the remaining firms continue to coordinate their production decisions. We further reinterpret the Cournot–Stackelberg hybrid structure as a natural punitive benchmark following cartel breakdown, in which the remaining firms abandon cooperation and compete while facing the deviating firm as a common Stackelberg follower. From a game-theoretic perspective, each firm’s quantity choice depends on its own action, its rivals’ actions, and the resulting market state. While simple inverse demand and cost functions permit equilibrium analysis by classical profit maximization, nonlinear specifications used in oligopoly research, such as isoelastic inverse demand, generally do not yield explicit reaction functions or closed-form equilibria. We therefore represent strategic adjustment as an iterative game generated by firms’ reaction mappings: at each stage, firms revise their production quantities in response to the current market configuration. Tripled fixed points of the resulting mapping characterize equilibrium and provide conditions for existence, uniqueness, stability, and convergence of successive quantity adjustments. Numerical comparisons, ranging from an illustrative triopoly to the asymptotic regime with many firms, characterize how the failed-cartel and punitive Cournot–Stackelberg structures differ from classical markets in terms of output, prices, profits, consumer surplus, and total welfare.

Cartels may partially collapse when one firm abandons a collusive agreement while the remaining firms continue to coordinate their production decisions. Such market structures naturally lead to coupled optimization problems in which firms’ optimal responses need not be unique. Motivated by this observation, we develop a general mathematical framework for failed-cartel models based on set-valued reaction and best-response correspondences. We first establish an abstract equilibrium existence theorem for three interacting optimization problems under continuity, quasi-concavity, and compactness assumptions. The proof combines Berge’s Maximum Theorem with the Kakutani–Fan–Glicksberg fixed-point theorem and does not require uniqueness of the firms’ optimal responses. The classical failed-cartel Cournot model is then obtained as a direct specialization by verifying that its payoff functions satisfy the hypotheses of the abstract result. For the single-valued case, we derive conditions under which the equilibrium is unique. We further investigate the influence of strategic interactions on equilibrium production. Besides the classical effect of heterogeneous production costs, we identify a purely strategic mechanism showing that, even under identical technologies, the ordering of equilibrium quantities is determined by the sign of the follower’s reaction to the leaders’ production decisions. Depending on this reaction, either the coordinated leaders or the deviating follower becomes the dominant producer. The proposed framework extends the classical analysis of partially coordinated oligopolies and provides a unified mathematical approach for studying equilibrium in failed-cartel markets with both single-valued and set-valued reaction mappings.